Smallest possible value of a + b + c

Dividing $ab = 432$ by $bc = 96$: $\frac{a}{c} = \frac{432}{96} = \frac{9}{2} \implies a = \frac{9}{2}c$

Since $a$ is an integer, $c$ must be an even integer. Given $c < 9$, the possible values for $c$ are $2, 4, 6, 8$.

Testing each case:

• If $c = 2 \implies a = 9, b = 48 \implies a + b + c = 9 + 48 + 2 = 59$
• If $c = 4 \implies a = 18, b = 24 \implies a + b + c = 18 + 24 + 4 = 46$
• If $c = 6 \implies a = 27, b = 16 \implies a + b + c = 27 + 16 + 6 = 49$
• If $c = 8 \implies a = 36, b = 12 \implies a + b + c = 36 + 12 + 8 = 56$

The smallest possible value of $a + b + c$ is 46.